Thread: Find two positive divergent sequences such that..

1. Find two positive divergent sequences such that..

Hello
Find $a_n$ and $b_n$ such that:
{ $an$} , { $b_n$} are divergent sequences
$a_n,b_n > 0$ for $n\geq1$
and { $a_n+b_n$ } is a convergent sequences.

The problem here is both must positive.
If not, $a_n=n$ and $b_n=-n$ is a nice counter-example.

But they must be positive.

2. Originally Posted by General
Hello
Find $a_n$ and $b_n$ such that:
{ $an$} , { $b_n$} are divergent sequences
$a_n,b_n > 0$ for $n\geq1$
and { $a_n+b_n$ } is a convergent sequences.

The problem here is both must positive.
If not, $a_n=n$ and $b_n=-n$ is a nice counter-example.

But they must be positive.

Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

Bigger hint;
Spoiler:
think sine and cosine.

3. Originally Posted by pomp
Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

Bigger hint;
Spoiler:
think sine and cosine.
But sine and cosine are not positive for all $n\geq1$

4. Originally Posted by General
But sine and cosine are not positive for all $n\geq1$
Well then do something to them that will ensure they are...

5. You mean $sin^2n$ and $cos^2n$ ?
What the limit of them ?
What the limit of thier sum ?
as $n \rightarrow \infty$

6. Originally Posted by General
You mean $sin^2n$ and $cos^2n$ ?
What the limit of them ?
What the limit of thier sum ?
as $n \rightarrow \infty$
"What the limit of them?"

Neither converge to a limit, they oscillate in (0,1)

"What is the limit of thier sum? "

What is $\sin^2 (x) + \cos^2(x)$ ?